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For example, I asked this Is $\{(x,y)|y=\sin(\frac{1}{x})\}\cup(0,1)$ connected on $R^2$?, in the answer someone points out that a similar question was asked and discussed before. The title looks different so I cannot find it by searching, should I delete questions like this myself( If reputation lost is a consideration, should I delete it before any answer is upvote?)?

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No, you shouldn't. Duplicates are generally useful (there are exceptional cases such as the 27th duplicate of $0.999999...=1$).

Also, you can't delete your question if it has an upvoted answer (or two answers, even without upvotes).

What you could do (once you became aware of duplication) was to flag your question for closure: flag $\to$ it is a duplicate. Does not matter now because there is already a vote to close.

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    $\begingroup$ Really? Two answers block deletion? Interesting. $\endgroup$ – Asaf Karagila Oct 30 '14 at 13:27
  • $\begingroup$ @Asaf see the semiofficial meta.stackexchange.com/a/5222 "When can't I delete" $\endgroup$ – quid Oct 30 '14 at 14:43
  • $\begingroup$ I guess one major upside to duplicates is that we are basically crowd-sourcing the task of thinking up every conceivable title for a commonly asked question. Although, that won't stop lazy posters from using gibberish titles (after they exhaust sensible ones)... $\endgroup$ – rschwieb Oct 31 '14 at 12:58

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